In supersymmetry algebra, $\mathcal{N}$ refers to $I=1,2,.. \mathcal{N} $ in $Q^{I}_{\alpha}$. My question is what does it mean to write $\mathcal{N}=(1,1)$ superalgebra?
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1$\begingroup$ See e.g. Wikipedia. $\endgroup$– Qmechanic ♦Commented Aug 1, 2023 at 14:35
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I think I figured out the meaning of this after some research so, I am posting an answer to my own question.
The answer is there is nothing called $\mathcal{N}=(1,1)$ superalgebra. The superalgebra is always named by $\mathcal{N}$ with integers.
The $\mathcal{N}=(1,1)$ actually means a supergravity multiplet so my original question was wrong. We get this multiplet as the massless level of IIA superstring theory. (1,1) stands for getting fields of two different chiralities.